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Let $\mathbb F_q$ be a finite field, where $q$ is an odd prime power. Let $R=\mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q+uv\mathbb F_q$ with $u^2=u,v^2=v,uv=vu$. In this paper, we study the algebraic structure of $(\theta, \Theta)$-cyclic codes of block length $(r,s )$ over $\mathbb{F}_qR.$ Specifically, we analyze the structure of these codes as left $R[x:\Theta]$-submodules of $\mathfrak{R}_{r,s} = \frac{\mathbb{F}_q[x:\theta]}{\langle x^r-1\rangle} \times \frac{R[x:\Theta]}{\langle x^s-1\rangle}$. Our investigation involves determining generator polynomials and minimal generating sets for this family of codes. Further, we discuss the algebraic structure of separable codes. A relationship between the generator polynomials of $(\theta, \Theta)$-cyclic codes over $\mathbb F_qR$ and their duals is established. Moreover, we calculate the generator polynomials of dual of $(\theta, \Theta)$-cyclic codes. As an application of our study, we provide a construction of quantum error-correcting codes (QECCs) from $(\theta, \Theta)$-cyclic codes of block length $(r,s)$ over $\mathbb{F}_qR$. We support our theoretical results with illustrative examples., Comment: 30 pages, 4 tables

In this paper, we introduce $\mathbb{Z}_{p^r}\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive cyclic codes for $r\leq s$. These codes can be identified as $\mathbb{Z}_{p^s}[x]$-submodules of $\mathbb{Z}_{p^r}[x]/\langle x^{\alpha}-1\rangle \times \mathbb{Z}_{p^r}[x]/\langle x^{\beta}-1\rangle\times \mathbb{Z}_{p^s}[x]/\langle x^{\gamma}-1\rangle$. We determine the generator polynomials and minimal generating sets for this family of codes. Some previous works has been done for the case $p=2$ with $r=s=1$, $r=s=2$, and $r=1,s=2$. However, we show that in these previous works the classification of these codes were incomplete and the statements in this paper complete such classification. We also discuss the structure of separable $\mathbb{Z}_{p^r}\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive cyclic codes and determine their generator polynomials. Further, we also study the duality of $\mathbb{Z}_{p^s}[x]$-submodules. As applications, we present some examples and construct some optimal binary codes.